If for some $\alpha$ and $\beta$ in $R,$ the intersection of the following three planes  $x+4 y-2 z=1$ ; $x+7 y-5 z=\beta$ ; $x+5 y+\alpha z=5$ is a line in $\mathrm{R}^{3},$ then $\alpha+\beta$ is equal to

Let the system of linear equations  $x+y+k z=2$ ;  $2 x+3 y-z=1$ ; $3 x+4 y+2 z=k$ , have infinitely many solutions. Then the system $( k +1) x +(2 k -1) y =7$ ; $(2 k +1) x +( k +5) y =10 \text { has : }$

If the system of linear equations  $x-2 y+z=-4 $   ;  $2 x+\alpha y+3 z=5 $  ;  $3 x-y+\beta z=3$ has infinitely many solutions, then $12 \alpha+13 \beta$ is equal to

If $a,b,c$ are respectively the ${p^{th}},{q^{th}}{r^{th}}$terms of an $A.P.,$ the $\left| {\,\begin{array}{*{20}{c}}a&p&1\\b&q&1\\c&r&1\end{array}\,} \right| = $

If the following system of linear equations

$2 x+y+z=5$

$x-y+z=3$

$x+y+a z=b$

has no solution, then :

If $\left| {\begin{array}{*{20}{c}}
  {^9{C_4}}&{^9{C_5}}&{^{10}{C_r}} \\ 
  {^{10}{C_6}}&{^{10}{C_7}}&{^{11}{C_{r + 2}}} \\ 
  {^{11}{C_8}}&{^{11}{C_9}}&{^{12}{C_{r + 4}}} 
\end{array}} \right| = 0$ then $r$ is equal to